Uniform H\"older regularity with small exponent in competition-fractional diffusion systems

TL;DR

This paper proves that for certain fractional competition-diffusion systems, uniform boundedness ensures H"older regularity with small exponents, leading to strong convergence as solutions become segregated.

Contribution

It establishes uniform H"older regularity results for fractional competition systems with small exponents, extending understanding of solution behavior as competition intensifies.

Findings

Uniform boundedness implies H"older boundedness for small exponents

Solutions converge strongly as segregation occurs

Results apply to fractional Gross-Pitaevskii system

Abstract

For a class of competition-diffusion nonlinear systems involving fractional powers of the Laplacian, including as a special case the fractional Gross-Pitaevskii system, we prove that uniform boundedness implies H\"older boundedness for sufficiently small positive exponents, uniformly as the interspecific competition parameter diverges. This implies strong convergence of the family of solutions as the segregation of their support occurs.